Sunday, March 27, 2016

Recently I was reading a biography of John Cage, the quirky avant-garde 20th-century classical composer who I have mentioned in a few previous podcasts. One of the most fascinating aspects of Cage’s composing was his attempt to introduce random elements into his music, starting in the 1950s, in order to free himself from preconceived patterns. He experimented with numerous sources of randomness, including die rolls, ambient noise from the environment, and even imperfections in the paper he was writing on. But one method that absorbed his interest for a long time was the ancient Chinese book of divination known as the I Ching. Once Cage discovered the I Ching, it became his main guide in the selection of random numbers. Some of his compositions required thousands of random numbers to be completed. As a result, many of his visitors noted that anyone who stepped through Cage’s door was soon drafted into tossing coins for a few hours to generate I Ching trigrams for use in Cage’s music.

The I Ching, or Book of Changes, is said to be one of the world’s oldest books, written around 3000 years ago. It is based around interpreting the significance of various patterns of whole and broken lines, traditionally determined by tossing yarrow sticks, or by an equivalent method based on tossing coins. The most fundamental set of patterns generated by the I Ching are the eight trigrams, patterns of three lines, which may each be solid or broken. Each of the eight trigrams has several possible meanings, such as the mind, the spirit, emotions, or bodily sensations. Pairs of these trigrams can be combined into one of 64 possible hexagrams, for an even richer set of possible meanings to interpret. Being a listener of this podcast, you have probably realized by now that the combinations of three or six lines, each of which can be solid or broken, is precisely equivalent to a three- or six- digit binary number, if you interpret the solid lines as 1s and the broken ones as 0s. So essentially, the I Ching is a divination system based on random numbers, expressed in binary, or base-2, notation, between 0 and 63. Now I’m sure Chinese scholars will say I’m shortchanging the deep philosophy of the system, since these random divinations are accompanied by thousands of pages of interpretive text. But it’s undeniable that these numbers are the basis.

Because of this numerical aspect, it’s actually not uncommon among historians of science to credit the ancient Chinese for first coming up with the idea of the binary number system, which is critical to modern computers. Personally I’m a bit skeptical of this aspect of I Ching studies: while the ancient book discussed many ways to combine and interpret the trigrams and hexagrams, they weren’t using these as a basis for a numerical system or for calculations of mathematical significance. On the other hand, the legendary Gottfried Leibniz, co-inventor of calculus and early designer of ideas for calculating machines, did credit the I Ching for inspiring the idea of binary arithmetic in some 17th-century writings. I think this may have been largely due to the fact that there were no other precedents for this idea in Leibniz’s time, though. Most likely, he was astonished to see some basic ideas of his base-2 new arithmetic system in this ancient text, though he probably would still have developed the binary system if unaware of these writings.

As I read more about the I Ching online though, I was surprised to see that its description as a system of binary numbers is actually a bit of an oversimplification. The reason is that the I Ching describes a complex procedure for generating the lines, not the simple 0/1 coin toss you would have guessed. When using the coin method to generate a solid or broken line, you are to toss three coins, with one side of each coin considered the “yin” side and the other the “yang” side. Each yin toss has a value of 2, which each yang toss has a value of 3. You then add the values together, to get a total between 6 and 9. A 6 or 8 is a broken line, while a 7 or 9 is a solid line. But there is more to it: the less probable 6 or 9 values indicate that their line is “moving”, while the 7 or 8 lines are “stable”. While the symbolic trigrams or hexagrams are still drawn with mainly solidness or brokenness visible, you need to note which are moving and which are stable, as this can make a major difference in the results of your divination. Thus, one might say that the I Ching is really a base-4 divination system rather than binary. In some of his writings, John Cage actually claimed to be using these stable and moving aspects to guide his randomly generated music.

But on top of the base-4 complication, there is yet one more mathematical wrinkle. While the totally random methods such as tossing sticks and coins are the most commonly used, one online scholar notes that the I Ching describes another, more complex, method for generating the next 6/7/8/9 line number based on the current one, using a series of mathematical calculations. These calculations are actually pseudo-random, similar to the Linear Congruential Generation algorithms used by modern computers. This means that the results are deterministic, though hard enough to predict that they appear random. Furthermore, according to this online analysis, the official I Ching algorithm is somewhat biased: while solid and broken lines are equally likely, the 9 is much more probable than the 6, meaning that solid lines are significantly more likely to be “moving” than broken ones. I’m sure New Age mystics would say there is some deep meaning in this, and that Yang is more mobile than Yin, or something like that. Being a bit more of a cynic, I would lean towards the interpretation that the ancient Chinese were just not mathematically advanced enough to notice the problem.

Anyway, I’m not sure how all this was supposed to lead to John Cage generating better music: while I really enjoy reading about his bizarre random methods, trying to listen to the resulting music for more than a few seconds at a time is not a very pleasant experience. It’s also amusing that Cage put so much energy into generating numbers using I Ching methods, when he could have bought books of pre-generated random numbers, which were available for engineering and cryptographic applications for decades before the advent of modern computers, and saved a lot of time. But I wonder if Cage’s avant-garde admirers would have claimed to like his music as much, if he told them the source was the Rand Corporation rather than ancient Chinese mysticism.

Friday, March 4, 2016

Hi everyone-- you may have noticed that a bunch of episodes recently disappeared from this blog & the corresponding podcast feed. Each of the disappeared episodes is included in the new Math Mutation book, which will be released in Spring 2016, and is already available for pre-order from Amazon.

Apologies to those of you who are searching for those episodes. I really wanted to keep them up on the web, but the publisher's contract requires their removal, out of fear that their free availability would hurt book sales. I tried to push back, but it's a standardized contract & there's not much wiggle room.

Note that the other episodes will remain on this site, and I will continue to release new ones (at least) monthly and post them as usual.

Note that this podcast is intended mainly for audio consumption, so you will not see the numerous illustrations & diagrams you would find at most math sites, though these are linked in the show notes whenever possible.

The Math Mutation Book

The Math Mutation book, "Math Mutation Classics: Exploring Interesting, Fun, and Weird Corners of Mathematics", is now available. You can order it from Amazon at this link.

By the way, if you like it, don't forget to post a review on Amazon. And of course, if you're in the Portland, Oregon metro area, I'll be happy to autograph your copy sometime.

Donations

If you enjoy this podcast enough to donate money, thanks! But since I don't spend an appreciable amount on creating it, I ask instead that you donate to your favorite charity, and send me an email about it. You will then experience fame and fortune as I mention your name on the next podcast.